David is making rice for his guests based on a recipe that requires rice, water, and a special blend of spice, where the rice-to-spice ratio is $15:1$. He currently has $40$ grams of the spice blend, and he can go buy more if necessary. He wants to make $10$ servings, where each serving has $75$ grams of rice. Overall, David spends $4.50$ dollars on rice. How many servings can David make with the current amount of spice he has?
Solution: There can be many ways to solve this problem. Here, we will do this by thinking about units. Let's say David can make $x\,\text{servings}$ using $40\,\text{grams}$ of spice. How can we relate these two quantities with an equation? $\begin{aligned} x\,\text{servings}\cdot y\,\dfrac{\text{grams of spice}}{\text{serving}}=40\,\text{grams of spice} \end{aligned}$ So in order to find the number of servings $x$, we need to figure out the value of $y$, which is the amount of spice per serving. Notice what other information we are given: $15\,\dfrac{\text{grams of rice}}{\text{gram of spice}}$ $10\,\text{servings}$ $75\,\dfrac{\text{grams of rice}}{\text{serving}}$ $4.50\,\text{dollars}$ (on rice) Which of these quantities can help us calculate a rate whose units are $\dfrac{\text{grams of spice}}{\text{serving}}$ ? We can combine the following quantities: $\begin{aligned} &\phantom{=}\dfrac{75\,\dfrac{\text{grams of rice}}{\text{serving}}}{15\,\dfrac{\text{grams of rice}}{\text{grams of spice}}} \\\\ &=\dfrac{75}{15}\,\dfrac{\cancel\text{grams of rice}}{\text{serving}}\cdot\dfrac{\text{grams of spice}}{\cancel\text{grams of rice}} \\\\ &=5\,\dfrac{\text{grams of spice}}{\text{serving}} \end{aligned}$ Now we can plug that in the original equation: $\begin{aligned} x\,\text{servings}\cdot 5\,\dfrac{\text{grams of spice}}{\text{serving}}&=40\,\text{grams of spice} \\\\ x\,\text{servings}&=\dfrac{40}{5}\,\cancel\text{grams of spice}\cdot\dfrac{\text{servings}}{\cancel\text{grams of spice}} \\\\ x\,\text{servings}&=8\,\text{servings} \end{aligned}$ In conclusion, David can make $8$ servings with the current amount of spice he has.